Re: Logarithm of transfinite numbers

imaginatorium@xxxxxxxxxxxxx said:

Tony Orlow wrote:
imaginatorium@xxxxxxxxxxxxx said:

Tony Orlow wrote:
imaginatorium@xxxxxxxxxxxxx said:

<snippino>

Anyway, please tell me a little more about the property of Orlovian
"unboundedness". I gather you make some claim that in your "system",
the set of finite naturals is "unbounded". Is that right?

Yes.


Can you then give me an example of a finite natural, in the set of
finite naturals, that is "unbounded"? I'm rather guessing that since
it's possible that we all mean the same thing by a "finite natural",
that is, something which can be represented by a string of (decimal)
digits, with two ends, and such that given enough time one could read
from one end to the other and stop, you will say that there are not any
specificiable, actual, finite naturals that have this property of
"unboundedness". Is that right?

Of course.

OK so far...

In particular, why does the following argument not show that the set of
finite naturals cannot be unbounded:

For every finite natural x in this set, it is true that the set of
finite naturals up to x is bounded, since after all, none of them is
more than x.

That is a property of the initial subset, and it is a property of the natural
that its initial subset is bounded by itself.

You seem to reiterate what I said. So that's ok.

This property relates to the set up to this point, and
what's more this is true for any value of x.

For any x you can identify, that x serves as a bound on all x up to that point.

Ah, well, now I really need to understand this "identify". (Or do you
just mean "any x I care to identify"?) If not, are you saying that
there are pofnats that cannot be identified? (Sounds odd, since we
agree they can each be written as a sequence of digits that starts and
stops, and could be read through by a human given enough time. Surely
this sequence of digits "identifies" the number, no?)

I mean that when you take any x, it is the bound on all smaller x, not that
there are naturals that cannopt be specified, except the largest, which is not
a specific number, but an internal bound.

Tony, I have suggested before that if you posted a lot less, you might
manage to think out a post carefully enough to make at least a tiny
amount of progress. The above sentence is far too long.

It's really hard to limit the number of posts without ignoring the questions
that are put to me, so I guess I'll have to start by ignoring this one. Or,
maybe I''l just ignore the suggestion.


Do we agree what the pofnats are: this term saves so much time compared
with having to write out the bit about "represented by a string of
(decimal) digits, having two ends, and such that given enough time one
could read from the left end and stop at the right end". My question is
whether the set of pofnats includes anything which "cannot be
specified". I do not understand how it could, since in the long
description of a pofnat I've essentially said that it can be specified
by this string of (decimal) digits.

The tail end of your sentence reads "except the largest...". Oh dear:
can you explain what this means?

If ...111 is the binary string of all 1's in all finite positions, this is your
largest finite, and is specified as far as it can be. The value can not be
specified in realtion to anything else.

I thought we once agreed that there is
no largest (pofnat)? So what does this part of the sentence mean? (And
I also have _no_ idea what an "internal bound" is.)


An internal bound means an bound which is a member of the set.




Tony's argument (1):
If a property of the set
relates to the set size up to a point, and says something is true for
all elements of the set, [[then that fact about the set size at every
single point in the set includes EVERY point in the set]], and pertains
to the set as a whole.

For each x there is a bounded subset, but there is no bound on the value of x
itself within S, so there no overall bound on the set. Boundedness is a matter
of a maximum value, which the overall set does not have.

Well, hang on. Does your argument (1) above work, or does it not? Or
does it work when you say it does, and not when you say it doesn't?
(You expect us to keep up with that?)

It works when it does. Boundedness is a bit different from finiteness.
Finiteness is a matter of quantity in my book, whereas boundedness is a matter
of definitive identifiable points beyond which the set does not cross. As you
add elements, their values increment with each one, but the boundedness is not
a measurable feature, but the mere existence of x asa bound. They're two
different types of properties.

I'm not aware of anywhere in mathematics where there can be an
"argument" stated in general terms, which works "sometimes".

No? Is it always the case that for x and y in R, x/y is in R? Is division by
zero allowed in standard mathemtics, or is it proscribed as an exception to
avoid results like "1=2"? There are all sorts of exceptions, most notably in
set theory itself, as a blanket exception to most of math.

That is
the whole point of abstraction: you show that such and such an
implication follows logically from the axioms you are using. A
mathematical result can never require an oracle to say whether or not
it "works" in a particular case.

The symbolic systems and their interpretations are an outgrowth and a
formalization of more concrete concepts concerning the universe. If your
abstractions are truly abstracted from these concrete concepts and done well,
then fine. If they are plucked out of the air without any fundamental
justification, then they are questionable, and if they produce nonsensical
results, then there is an excellent chance that they are simply wrong. If you
simply accept symbolic statements without understanding the underlying
justification for those rules, then you aren't doing yourself any favoprs in
the long run.


<sorry, can't be bothered with any more. Write less; think more>

Chide less, respond logically more. I guess you didn't want to respond to that
part, but that's par for the course around here.


Brian Chandler
http://imaginatorium.org




--
Smiles,

Tony
.

Relevant Pages

  • Re: Logarithm of transfinite numbers
    ... the set of finite naturals is "unbounded". ... Boundedness is a bit different from finiteness. ... I wonder if you are aware, that this "unboundedness" you are having ... like what everyone else calls "infinite", ...
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  • Re: Logarithm of transfinite numbers
    ... sum can represent an infinite value. ... the sum over all finite bits is finite. ... All you have in the set are finite naturals, ... "unboundedness". ...
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  • Re: Logarithm of transfinite numbers
    ... the set of finite naturals is "unbounded". ... The existence of the bounded subset for each x is true for every x. ... I wonder if you are aware, that this "unboundedness" you are having ... not some property of boundedness ...
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  • Re: Logarithm of transfinite numbers
    ... the set of finite naturals is "unbounded". ... "unboundedness". ... Do we agree what the pofnats are: this term saves so much time compared ... Boundedness is a bit different from finiteness. ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... sum can represent an infinite value. ... the sum over all finite bits is finite. ... All you have in the set are finite naturals, ... not some property of boundedness ...
    (sci.math)